Vol. I V No. 4

GROUP THEORETICAL METHODS IN THE OPTICS*

Por: Kurt Bernardo Wolf'

* Contribution presented to the Seminar'on Group Theoretical Methods in Phisiscs. USSR o f Sciences, Yurrnala, Latvian SSR May'85.

+ Permanent Address: Instituto de Investigaciones en Matemá- ticas Aplicadas y en Sistemas (IIMAS), Universidad Nacio-- nal Autónoma de México, Apdo. Postal 20-726, O1000 México D.F., TELEX 1764062 IMASME.

DEPARTAMENTO DE MATEMATICAS

AV. MICHOACAN Y PURISIMA, COL. VICENTINA, IZTAPALAPA MEXICO, D.F., C0.P. 09340 APDO. POSTAL 55534

UNIVERSIDAD AUTONOMA METROPOLITANA-IZTAPALAPA

May 1985

A B S T R A C T

We sketch the results and lines of enquiry being pursued

in the application of Lie algebraic and group theoretic

methods for the study of optics. The hamiltonian formu-

lation leads to the optical phase space, where 3 ganss-

ian and a euclidean models spply. Free propagation and

refracting surface transformations concatenate to optical

systems -which may also be inhomogeneous or fiber-like.

These are represented by elements of a finite-parameter

aberration order Lie group, acting on chosen phase- Jh

spaces for the geometric and wave-optical cases. Symme-

tries of refracting surfaces are described. Nonlinear

transformations of these spaces are the product of aberra-

tions of the system, which we treat fully to third order,

and refer to results on higher orders. Some open ques-

tions are suggested.

1

Optics has served as the expermental basis and model for much

of science since Galielo. The minds at work in mechanics were so too

in optics, in its early geometric aspects through Snell up to Newton,

and its wave aspects since Huygens through Fourier and including Som-

merfeld. Hamilton's original application of generating functions for

canonical transformations was in optics; only later were they recogni-

zed as a sharp tool for abstract mechanics through the introduction of

the concept of a phase space with a symplectic metric!) This concept,

in embrionary form, served Heisenberg') and Wey13) for the introduction

of quantum mechanics, where the Hamiltonian is quadratic in momentum.

The history and scope of Sophus Lie's work.is well known to this au-

dience, as is the fact that its manifold applications have been most

successful when apPlied to systems with quadratic Hamiltonians!) Al-

most invariably, the kinetic energy term is p /2m. This Hamiltonian

is indeed also useful in optics, albeit in its Gaussian approximation,

i.e., first-order optics which deals with linear transformations of

phase space. The associated methods have been under researcn since

the early seventies from the perspective of quantum mechanics5) and

groups of integral transforms.

2

6,71

It came to me as a pleasant surprise, upon reading the work

of Nazarathy and Shamir:) that the theory of canonical transforms had

a very natural model in paraxial wave-optical systems. The elements

of such systems are usually slabs of homogeneous material separated

by 'almost flat' refracting surfaces. There is an association be-

tween these optical elements and elements of a group with the struc-

ture W Sp (2N, R) , where Sp (2N,R) is the symplectic group of linear - transfornations of 2N-dimensional phase space, in semidirect product

with W N , the N-dimensional Heisenberg-Weyl group of phase-space trans-

lations with a one-dimensional center.

N

From a different direction, the work of Alex Dragt’) and

coworkers lo”l) in nonlinear orbit analysis for accelerators and ’- --

electron microscopes, was based on the theory-of -Lie seriesAL’A4’

applied to classical phase-space variables. This allows the cal-

culation -by electronic computer- of particle or light ray tra-

jectories up to an in principle arbitrarily high aberration order.

Upper practical limits for this order are seven or nine. MARYLIE

e ..,,

14 1

is a program which models magnetic-lens optical elements, concate-

nates them into an operator description of the system, and applies

the latter to the object space to produce the image space result

of the system. It works in three dimensions in coordinate space,

since magnetic elements are in general not axially-symmetric they

need two transverse coordinates, and one z-axis along which chroma-

tic dispersion of the electron packet occurs. This calls for Sp(6,R).

The phase-space translation group, W , is involved, moreover, in a very direct way in radar signal analysis.

cause the aim is to determine the distance through the echo time-lag

(space translation) and the target velocity through the frequency-

shift (momentum translation). The optimum resolution problem, for

instance, has been solved using harmonic oscillator functions. The

group of linear transformations pertains the-detection of accelera-

tion, and higher-order aberrations to higher-order derivatives of

velocity.

N15, 16,17) This is be-

The bag of nearby applications of Lie optics grows with the

orderly account it.makes of coherent states in their behavior

in optical fibers with inhomogeneities and interfaces, and the measu-

rability of phases in microwave systems. This should be germane to

the design of optical computers. As it will become clear at the end

of this paper, the purely group-theoretical aspects of optical systems,

or any other means of image processing, are also of interest in them-

selves.

18,19)

3

Section 2 contains a brief recount of the hamiltonian for-

mulation of free-flight optics and Section 3 of refracting surfaces. ?

Their concatenation in Section 4 is done through Lie group multipli- . . ~ . . .

cation. We present the symplectic classification of higher-order

aberrations, in particular of aberrations of order three. In Sec-

tion 5 we outline their effect on the phase space of geometrical

optics and on wavefunctions in wave optics. The handling of inho-

mogeneous systems is sketched in Section 6 , and Section 7 we offer

some open questions.

2 . The hamiltonian formulation

The application of Fermat’s minimal action principle leads

to the hamiltonian formulation of optics. The system is given a

reference optical axis z and, in a z=constant plane perpendicular

to this axis, we set our coordinate system q. This ’position’ spa-

ce is two-dimensional for actual optical systems, but is reduced to

one dimension if further symmetries exist (cylindrical and axis-sym-

metric lenses). A light ray crossing the coordinate plane at 5 has its direction given by a second coordinate system of angular varia-

bles. If n(q,z) is the value of the refraction index at the point

( a , z ) , and n is a vector of magnitude n in the z=constant plane, the projection of the ray on the plane, then the coordinate canonically

conjugate to 9 is shown 9‘20) to be

-

-

where 0 is the angle betiween the ray and the optical axis. In these

terms, the optical Hamiltonian is

4

. From this starting point, the well-known hamiltonian machi-

nery21) defines Poisson brackets, equations of motion, canonical - ~.

A. c L A . .

Lie algebras. 22) Free propagation F in a z-homogeneous medium, in

particular, belongs to the one-parameter evolution group

/..

2

A

Equation (3a) is the usual evolution-operator construction which for

(2) yields (3b), where the first-order or Gaussian terms are made ex-

plicit in matrix form. The Gaussian part is readily wavized when we

replace the matrices (finite-dimensional non-unitary representations

of S p ( 2 , R ) ) by integral canonical transforms (infinite-dimen-

sional unitary representations of the double cover symplectic spin-

of the same group).

We should be careful in noting that this treatment of Gauss-

ian optics, and that of optical aberrations,which will be presented

below, disregards the fact that the geometry of optical phase space

is not that of a plane, but that, due to (1) and (21, has momentum

coordinates which range over a sphere. For N = l dimension, this has

been taken into account in a manner very similar to that of quantum

mechanics on a compact space (a circle S 1 in Ref. 23. We have then

a mixed Heisenberg-Weyl group, with two infinitesimal generators, Q =

i?íd/de and I = 5 ; (where 2 = 3 /2sfis the reduced wavelenght in the system), and one finite generator E (which in the Schrddinger case is

exp iP), whose action is multiplication by exp ie. The two former ge-

-

1 h

h

.4

A

nerators are self-adjoint in L (S1) and the latter is unitary. Out of

this one, nevertheless, we may build the two self-adjoint combinations Ir - ’t P := n (E-E )/21 and f; := -n (;+2’)/2, which in Ref. 24 are identified

with the momentum and Hamiltonian, (1) and ( 2 ) . Perhaps surprisingly,

L

5

e

these three operators ( Q , P, and HI c lose in to a euclidean iso(2)

algebra. For paraxial optics, the matrix elements between forward-

concentrated wavefunctions2') o f t h e generators of t h i s euclidean

a lgebra contrac t to the u s u a l Heisenberg-Weyl algebra W Free pro-

pagation belongs t o a one-parmeter subgroup o f the associated Lie

f i n

" ~ .~

1'

group ISO(21, which i s t h u s the dynamical 4 ) group o f free-propaga-

t ion opt i cs . One o f the consequences o f t h i s construction, however,

i s t h a t configuration space must be d i s c r e t e , with points separated

by x This i s q u i t e reasonable i n view of the Whittaker-Shannon

sampling theoremZ5) The propagator i s given by a Bessel function:

The system i s thus re lated to that of an i n f i n i t e l a t t i c e o f masses

joined by springs?6f (The l a t t i c e Hamiltonian i s P instead of H as

here , i . e . the system i s 'rotated' by g o o . )

h A.

Fundamental Gaussian functions for the system are defined 24)

as diffused Dirac d e l t a s , G w ( q ) := Fiw (k=q/% ,O) =, Ik ( w n ) , and given

i n terms o f the modified Bessel functions which, indeed, look Gauss-

i a n . F i n a l l y , we may deform the euclidean algebra (where P2 + H 2 =

n ) t o s o ( 2 ,I) = sp(2 ,R) or to other2*) inf ini te-dimensional a lgebras . 2

The s t a t e of t h i s s tudy i s preliminary, however. More than

delving into t h e fundamentals o f t h e hamiltonian description o f o p t i c s ,

here we would l i k e t o apply the well-tested Schrddinger formalism on

a f l a t phase space, i .e. on L (R i n s p i t e o f some o f i t s d i f f i c u l -

t i e s , s i n c e t h e aim i s t o c a l c u l a t e e f f i c ient ly the behaviour of

sp(2,RI-defined objects under increasing aberration order. I n R e f . 21

we pursued t h i s ob ject ive for the study o f Gaussian beams -coherent,

d i s c r e t e , and correlated- under the aberration of free propagation.

2 N

6

3 . Refracting surfaces

- ..~_ .~ .

Optics contains one very basic phenomenon which does not ". ~ -

have a counterpart in mechanics: refracting surfaces. The clo-

sest mechanical model is that of a sudden change in the potential,

but the truth is that the latter is by force continuous 1) . Besides,

the instant of change of the potential should depend on the posi-

tion, if curved surfaces z = y(2) are to be subject to analogy. Refracting surfaces may be seen better as 'sudden' finite trans-

formations of phase space which take place at some reference plane

(made to conicide with the optical center $ ( O ) = O for convenience).

Thus, we search to define an operator s(n,n';s) accounting for the

refraction transformation a t 5 , between media n and n'. when the

two media are homogeneous, we showed in Ref. 28 a rather neat re-

sult. First, this transformation may be factorised as

- h

Second, the root transformation g(n; 5 ) may be written as

Third, this root transformation is canonical; it is an implicitly de-

fined transformation, since n appears on both sides of (5c) but may ,

be self-replaced by symbolic computer algorithms2*) to any desired

aberration order. Fourth, the above mapping is not globally One-tO-

one, but must have caustic singularities. _L_

The validity of this factorisation property for surfaces of

discontinuity in the refraction index has been shown29) to extend to

the case where 3 is the interface between two continuous, inhomoge-

neous media n(q,z) and n'(g.,z). This seems to be thus a rather fun-

damental property of things refracting worthy of deeper inquiry.

7

Refracting Surfaces, moreover, posess invariants. This is

obvious in the form of Snell ‘ S law:

V I 3 I 3 1 v \ y \ w cu * _ _ \ p : /f I - I (b 1

This Snell (vector) invariant has one value in medium n, and the same

value in medium n’; it is written in terms of the root-transformed

variables 6 and in ( 5 ) . In fact, any function of E and 9 alone is

an invariant. The Snell invariant in two dimensions has the follow-

ing interesting property:

- -

4 \ ”

Y2 m The last expression is

(spherical, Y,

surf ace 1 (8c)

the well-known skewness or Petzval invariant 31,32) of an axis-symme-

tric surface. For a spherical surface, we see above, we have an S o ( 3 )

algebra of invariants which can be proven, predictably enough, to be

the generators of phase-space transZormations which stem from rota-

tions in space which leave the surface invariant.

The Snell invariants allow for an economical description of ~

aplanatic point pairs32) and -for spherical surfaces- is explicitly

expressible as a function of the object and image coordinates as

8

Through series expansion 2’ ( ~ ~ 9 ) , z’ C E , ~ ) , this formula allows the recursive computation of the expansion coefficients by comparison :- ~

ot the Independent powers ot E and z. ’I’nls algorltnm has Deen pur-

sued to ninth order 30’31) to yield the analytic expressions for the

aberration coefficients (to be detailed below) of a Spherical sur-

face. Certain ‘selection rules’ which determine zeros f o r some

coefficients (generalized spherical aberration, coma, and astiqma-

tism) , which have been observed by D r a ~ g t ~ ~ ) and Forest‘’) are explai-

ned in terms of optical center conditions of the surface. Similar

analyses should be applicable to other revolution conics.

4 . Concatenation of optical elements

e-

Having described homogeneous free-space propagation and re-

fracting surfaces, we concatenate them to form lens optical systems.

Here we have to use the enveloping algebra of the classical Poisson-

bracket Heisenberg-Key1 algebra. Tnere is a nesting in this infinite-

dimensional algebra, however, by polynomial order. 35) The Poisson brac-

ket between an Nth order homogeneous polynomial in E and 5, and an N‘ th

order one, is of order N + N’ - 2. Second-order polynomials thus close

into sp(2,R). Under this, Nth order polynomials transform (by adjoint

action) as an ideal. Since polynomials of order larger than N+1> 2

form an (infinite-dimemional) subalgebra, we may build the factor al-

gebra of the full enveloping algebra modulo the latter. 36)’ This is the

(finite-dimensional) Nth-order aberration algebra. It is generated by

the quadratic monomials ”

and by monomials hx: of order k = 3, 4 , .. . , N+1. The Poisson brackets among the latter are set congruent with zero whenever the order exceeds

N + 1.

9

The group-theoretical classification of aberrations 30,36,37)

is done in terms of finite-dimensional (non-hermitian) irreducible .-

Y% 1 1 1 )

When the optical system has a common axis of rotational symmetry (the

optical axis), the aberration algebra will contain only monomials of

even order in E and T, being thus a function of p , p-q, and q only.

We may then define the polynomials kx, k=2,4,6,. . , j = k / 2 , k / 2 - 2 , . . . , 1 or O, m=j,j-1, ...,-j, as homogeneous polynomials of order k, spin j,

and projection m. The last label is the eigenvalue under the sp(2,R)

weight operator (p-q/2) . These symplectic-classified functions can

be written in terms of the ordinary solid spherical harmonics j g ~ 1 in the angular variables

2 2 - 1

"

-

with invariant radius

which is related to the Petzval invariant. The expression for the

maximum- j polynomials is 30 1

For the less-than-maximum j polynomials, they are obtained through multiplication of the order-four invariant (13) , to form

The elements of tne (2n-lIth order aberration algebra may be thus

written as

To simplify matters, let us work henceforth in third aberra- - tion order, which yields compact, explicit results. Hiqher orders

are as yet under development to find the Berezin bracket38) for the

sp(2 ,R) -based phase space description. By use of REDUCE3’) symbolic

computation algorithms, Miguel Navarro Saad has developed4’) various

programs which are being integrated as an interactive master program

for the design of optical lens systems to aberration order up to nine;

the expressions are all analytic and may be printed in explicit form,

spot diagrams may be generated, and other forms of graphic display

are being contemplated. Group-theoretically, the bracket algorithm

is still incomplete.

For aberration order three:6) the generators of the aberration

algebra of third order are (11) and those of fourth order: the symplec-

tic quintuplet is

and the symplectic singlet,

(valid for all N and j integer, and j half integer as well).

11

The Lie group corresponding to the aberration Lie algebra seen

above is the 3 -order aberration qrouD. We choose its presentation as rd -

where M(n) is a 2 r 2 unimodular matrix, v a 5-vector, and w a scalar.

The group multiplication law is ” -

irreducible representation of M E Sp ( 2 ,R) , - ”-

(also valid for half-integer j). Now, the group elements corresponding

to the optical elements of the lens system are:

Free proFaaation, given by the one-parameter subgroup line

An up-to-quartic refracting interface

for rays passing from a medium n to another n’, is associated to

The names of the third-order (Seidel) aberrations are: 34) va spheri-

cal aberration; vl, coma; C=v =3w/4, curvature of field; A=v -3W/2,

astigmatism; and vq1, distortion. O O

12

Suggested by group theory are: 36) v curvatism; w, astigmature;

and the last (not counted by Seidel) aberration, v dubbed pocus, =

O '

-2 e k.ourier conyugate of spherical aberration, and ac-

counts for p"unfocusing; this produces a decreased depth of field

for large angles.

"-

To concatenate these elements, we place our optical ele-

ments from left to right, write under them their corresponding

group elements ( 2 3 ) and ( 2 5 1 , and multiply using ( 2 1 ) . The alge-

bra is manageable by hand for up to around five optical elements.

Thereafter, as well as €or higher aberration orders, we resort to

s;.mbolic computation. The aberration coefficients €or eighth and

tenth-order surfaces have been obtained by Forest'') and Navarro-

Saad. We note that, due to this symplectic-multiplet classifica-

tion, the six third-order aberrations transform under Gaussian op-

tics through a block diagonal matrix (5+1=6). For higher aberra-

tion orders this reduction of computation may be significant. For

aberration order five, for example, dimensionalities are 7+3=10;

for orden seven, 9+5+1=15; and for nine, 11+7+3=21. The compoun-

ding of aberration polynomials under multiplication, € o r orders

higher than third, however, is still under development. Perhaps

predicably, Clebsch-Gordan coefficients and triangle selection ru-

les appear.

5. Aberrations of phase space

We have the aberration group elements and their multiplica-

tion. The objects on which this group acts depends on the realiza-

tion of optics we want. In terms of the geometrical phase-space ope-

rators introduced in the second section, we must act with the expo-

nential series13) in ( 2 0 ) on E and 3, thereby generating nonlinear terms for j yl,; these terms may be molded into syinplectic polyno-

13

mials which may be classified in a way similar to (141, but in

terms of sp(2,R) spinor spherical harmonics. For aberration or- ,- c 3 - rv: UL ..J. . "-S, 2s

may be expected, introduce the sp (2 ,R) spinor E , each of two so(2)-components, coupled with Clebsch-Gordan coefficients to the

harmonics ( 1 4 ) . In this way one produces a canonical map, the

map is nonlinear as E' (e,%) , 9' (E,%) , and linear in the full ba-

( 5 )

In references 36 and 41, we have cut the phase-space coor- - dinates after the term with the aberration order. This means for

aberration order three to have the two linear terms

a symplectic quartet x:'2 and a doublet x, ; the last six

polynonials are cubic. We cut after that, since the essential

group action is contained in this eight-dimensional space. The

representation matrix in this space is minimal in dimension, and

contains linear transformations in ~ - 2 space, linear transforma-

tions mong the six cubic terms (decomposed into a quadruplet and

doublet), and finally, mixing cubic terms into the linear ones,

a 2 x 4 off-diagonal submatrix, and a scalar. We need not present

the details here, since they appear in the references, but only re-

mark that spot diagrams €or systems or for individual aberrations

up to third order appear in reference 28.

1 1/2 Km = E,9, 3 1/2

T-.e aberration group to third order has nine parameters,

three GaILssian ones and six aberration parameters. This group may

be made LO act on other spaces to model wave optics. At the outset

we should ward the reader that this matter is not fully controlled

yet, but several open avenues and a few caution marks will be sketch-

ed below.

The SchrMinger approach consists in using L (R) functions 2

and the usual realization of the canonical generators p, f), in pla-

ce of the classical Poisson generators G , 4, replaced into the aber-

14

ration polynomials (14). Indeed, this works for the Gaussian part.

For the aberration part, involving the exponentials of fourth- and :;.

nlgher - oraer poAynomlals, cnls 1s not so stralgntrorwara. ~ . .

higher-order polynomials are not ~elf-adjoint~~) in L 2 (R) , nor do

42 1

they have a unique q~antization~~) ; P2Q2 and (P.*)* fail on both ac-

counts. The first objection may be circumvented working with matrix

L (Sl) :4) as we did in the second section. 2 elements between forward-concentrated beams:') or by working within

For the latter, the uni-

queness-of-quantization problem, we should

tion rule, since only42) by this rule will

to hold.

It does not seem to be possible to

demand the Weyl quantiza-

equations (19) continue

build an integral transform

realization of the finite-parameter aberration group, which would act

as a Huygens-Fresnel integral to third order. Nevertheless, we may

try to cut the space in some way if we consider only the first power

term in the exponential series of the aberration part. this means to

act with differential operators of fourth order for spherical aberra-

tion, third-order €or coma, etc., in Weyl order. Very preliminary re-

sults indicate44) that when these operators are made to act on Gaussian

spot functions, the resulting pattern indeed models the known diffrac-

tion patterns in a5erration (Ref. 32, Fig. 9.61, for small coefficients.

The effect on Gaussian spots of going to higher orders in the cut-off

of the exponential series of the fourth-order terms has been started

in collaboration with Wolfgang Lassner (NTZ, Leipzig), but are not yet

available. (Our computer at IIMAS had a breakdown -a real crash!)

-

A s may be gathered from the discussion in this section, the al-

gorit'ms provided by Lie methods and those by traditional ray tracing,

are quite different. Once the representing group element has been ob-

tained, it serves for any ray in phase space (or outside it, if we count

on it to be in S I ) , and may be subject to a wave-optics translation. As

we now turn to describe inhomogeneous-space optics, the effect of this

difference may be appreciated.

the optical system inhomogeneous. The first case includes models

for optical fibers with a quartic index profile

A Hamiltonian depending on z will generate a line which twists with-

in the group, and is not coincident with a subgroup. Our results here are explicit only to third aberration order. 41) Higher orders should

follow suit, but there aberrations compound.

The tangent to a line in the group G

given by

4

When the system is purely %-inhomogeneous, as fibers are, the

Hamiltonian stemming from (26) has the form

16

This general exponential relation is c lo sed , and so l ve s ana l4 t i ca l l y

every Baker-Campbell-Hausdorff relat ion within the group. Part ia l

resu l t s ex i s t for aberra t ion orders h igher than th i rd , and cer ta in

p a r t i c u l a r c a s e s ( s u c h a s e l l i p t i c 13kI produced from (28) fo r f i ber

o p t i c s , and t r iangular M a s from refract ing surfaces) have been so l -

ved, the general ca se remains to be worked o u t .

- -

The case for quar t i c f i ber s (26) has been g i v e n e x p l i c i t l y i n

Ref. 41, and shown tha t a l l aberra t ions bu t a s t i gmature may be made

to van i sh by appropriate choices of the rat ios n :$:f and the f iber

lenght z . The ca l cu la t ion i s performed through a Bargmann matrix

transform B(u) in (29d) which diagonalizes the Gaussian part of the

system. A detai led s tudy of th is system, with experimental predict ions ,

however, remains t o be performed.

O

”

17

7 . O u t l o o k

Only the barest essentials of the current program on Lie -

optics have been outlined here, and much remains to be done. Care-

ful comparison of cost with existing systems from more traditional

methods of ray tracing and Fourier optics must be performed before

we can claim true improvement in this old science. The work of

Dragt 9’14) on accelerator design indicates this is probably the

case. ?resently, with Navarro Saad, we are developing spot dia-

grams for spherical surfaces, where aberration orders 3 , 5, 7 , and

9 may be compared with the exact result over the full surface. The

movement of Gaussian coherent .beams within fibers46) with inhomo-

geneous necks or kinks4’) is also being readied for graphic display, 48 1

as well as their travel under free-space spherical aberration. This

problem has recently been examined together with V.I. Man’ko. 20 1

We should note, as an unsatisfactory feature of the ‘Schrbdin-

ger’ description, that only wavefront propagation to the right is des-

cribed. Reflection phenomena, partial and total, must be accounted

for, probably through an 6 la Dirac4’) particle-antigarticle space

doubling. This should take into account the two signs of the root

function in the Hamiltonian. The off-diagonal terms are expected to

involve the refraction index gradient or discontinuity.

Finally, the more fundamental description of optical phase-

space and its associated euclidean Heisenberg-Weyl group may yet offer

some pleasant surprises when applied to aberration optics. I should

offer my apologies for showing only the scaffold of these unfinished

matters.

Acnowledgement. I would like to thank the organizers of the Yurmala

Seminar for inviting me to present this contribution.

18

R E F E R E N C E S

"-

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I.A. Malkin and V . I . Man'ko, Dynamical symmetries and coherent states of quantum systems. (Nauka, MOSCOW, 1979) , in Russian. M. Moshinsky and C. Quesne, Oscillator systems, in: Proceedings of the 15th Solvay Conference in Physics (1970). (Gordon and Breach, 19741, ib. Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772-1780 (19711, ib. Canonical transformations and matrix elements. J. Math. Phys.

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- 12 , 1780-1783 (1971). K.B. Wolf, Canonical transforms I. Complex linear transforms. J. Math. Phys. 15, 1295-1301 (1974)ibid. 11. Complex radial trans- forms. J. Math Phys. 15,2101-2111 (1974)ibid. IV. Hyperbolic trans- forms : continuous serGs representations of sl(2 ,R) . J. Math. Phys. - 21 680-688 (1 9 8 0 ) . K.B. Wolf, Integral transforms in science and engineering (Plenum Publ. C o r p . , 1979) , part IV. M. Nazarathy and J. Shamir, Fourier optics described by operator algebra. J. Opt. Soc. Am. 70, 150-158 (1980); ib. First-order op- tics a canonical operator representation:^ lossless systems. J.

- Opt. SOC. Am. 72, 356-364 (1982) . - A.J. Dragt, Lectures on nonlinear orbit dynamics. (AIP Conf_erence Proceedings, Vol. 87, 1982). -

10) D.R. Douglas, Lie algebraic methods for particle accelerator theory. Ph. D. Thesis, University of Maryland, 1982.

11) E. Forest, Lie algebraic methods for charged particle beams and light optics. Ph.D. Thesis, University of Maryland, 1984.

12) A.J. Dragt and J.M. Finn, Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17, 2215-2227 (1976). -

13) S. Steinberg, Factored product expansions of solutions of nonlinear differential equations. SIAM J. Math. Anal. 15, 108-115 (1984).

14) D.R. Douglas and A.J. Dragt, MARYLIE, the Maryland Lie algebraic -

beam transport and particle tracking program. IEEE Trans. Nucl. Sci. NS-30, 2442 (1983).

19

is) J.R. Klauder, Bell Syst. Techn. J. 39, 809 (1960); R . J . Glauber, - Phys. Rev. Lett. 10, 84 (1963). -

- _ ~" . ~ ~. LOI w. scnempp, maar reception and nilpotent nannonic analysis.-I-V.

C. R. Math. Rep. Acad. Sci. Canada 4, 43-48, 139-144, 219-255, 287-292 (1982) ; ibid. 5 . 5 , 3-8, 35r40 (1983) ; H. Raszillier and W. Schempp, Fourier optics from the perspective of the Heisenberg group. Preprint Univ. Bonn HE 84-34, to appear in Lie methods in optics, Proceedings of the CIFMO-CIO workshop on- Jan 7-10, 1985 León', Mexico, ,ed. by J. Sánchez Mondragón and K.B. Wolf; Lecture Notes in Physics, Springer Verlag.

"

17) M. Schmidt, Die reele Heisenberg-Gruppe und einige ihrer Anwendun- gen in Fadarortung und Physik. Diplomarbeit Universitttt-Hochshule Siegen, DBR, April 1985.

18) V.V. Dodonov, E.V. Kurmyshev, and V.I. Man'ko, Phys. Lett. 79A, 150 (1980) . -

. . . . .

19) J . R . Klauder and E.CGSudarshan, Fundamentals of quantum optics. . .

(Benjamin, 1968). 20) V.I. Man'ko and K.B. Wolf, The influence of aberrations in the op-

tics of caussian beam propagation. Preprint Univ. Metropolitana (April 1985); to appear in shortened version in Lie methods in OP- tics, op. cit. -

21) See e.g., H. Goldstein, Classical mechanics (Addison Wesley, 1959).

22) P. Exner, M. HavlíEek, and M. Lassner, Canonical realizations of classical algebras. Czech. J. Phys. B26, 1213-1228 (1976); W . Lass- ner, Noncommutiative algebras prepared for computer calculations. In: Proceedings of the International Conference on Systems and Tech- niques of Analytical Computing and Their Applications in Theoretical Physics. Dubna report Dll-80-13.

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23) H. Bacry and M. Cadilhac, Metaplectic group and Fourier optics. Phys. Rev. A. 23, 2533-2536 (1981). -

24) K.B. Wolf, The Heisenberg-Weyl ring in quantum mechanics. In: Group theory and its applications, Vol. 3, ed. by E.M. Loebl (Academic Press, 1975) . Section VI.

24) K.B. Wolf, A euclidean algebra of hamiltonian observables in Lie optics. Preprint, Universidad Metropolitana. May 1985.

25) J.W. Goodman, Introduction to Fourier optics. (Mc Graw-Hill, 1968)' Section 2.3.

26) Ref. 7, Sect. 5 .3 .

2 7 ) M. Navarro-Saad and K . B . Wolf, Factorization of the phase-space trans- fonnation produced by an arbitrary refracting surface. Preprint CIN- VESTAV (April 19841, to appear in J. Opt. Soc. Am.

20

28) M. Nabarro-Saad, Cálculo de aberraciones en sistemas ópticos con teoría de grupos. Tesis, Facultad de Ciencias, Universidad Nacio- nal Autónoma de México, Mexico DF, Jan. 1985.

c

A.J. Dragt, E. Forest, and K.B. Wolf, to appear in Lie methods in Optics, op. cit.

O.N..Stavroudis, The optics of rays, wavefronts, and caustics. (Aca- demic Press, 1972.)

M. Born and E. Wolf, Principles of optics. (Pergamon Press, 1959.)

M. Navarro-Saad and K.B. Wolf, Applications of a factorization theo- rem for ninth-order aberration optics. Comunicaciones Técnicas IIMAS preprint Desarrollo No. 41 (Feb. 19851, to appear in Journal of Sym- bolic Computation.

A.J. Dragt, Lie-algebraic theory of geometrical optics and optical aberrations. J. Opt. Soc. Am. 72, 372-379 (1982). - K.B. Wolf, Approximate canonical transformations and the treatment of aberrations. One-dimensional simple Nth order aberrations in optical systems. Comunicaciones Técnicas IIMAS preprint No. 352, 1983.

M. Navarro-Saad and K.B. Wolf, The group-theoretical treatment of aberraring systems. I. Aligned lens systems in third aberration or- der. Comunicaciones Técnicas Ii3lAS preprint No. 363, 1984.

H.A. Buchdahl, Optical aberration coefficients (Oxford Univ., 1954.)

F.A. Bereain, The method of second quantization. (Nauka, 1965.) In Russian . A.C. Hearn, REDUCE-2 User's Manual. University of Utah.

Ref. 28 and work in progress.

41) K.B. Wolf, The group-theoretical treatment of aberrating systems. 11. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order. Comunicaciones Técnicas 1U.IA.S preprint No. 366 1984.

42) M. García-Bullé, W. Lassner,and K.B. Wolf, The metaplectic group within the Heisenberg-Weyl ring. Universidad bletropolitana pre- print - 11, No. 20 (January 1985).

43) J.R. Klauder, Wave theory of imaging systems. To appear in Lie me- thods in optics, op. cit.

44) K - B . Wolf, work in progress, to appear in Lie methods in optics, op.cit.

45) G . W . Farnell, Canadian J. Phys. 36, 935 (19581, Figs. 3. -

21

46) K.B. Wolf, On time-dependent quadratic quantum Hamiltonians. SIAM J. Appl. Math. 40, 419-431 (1981). -

. - ." 471-0. Castaños, E. L6pez-Moren0, and K.B. Wolf, The qroup-theore- tical formulation of qaussian optics. Work in progress.

48) J.F. Barral, O. Castaños, S. Cuevas,and K . B . Wolf, work in pro-

49) J. Plebaiiski, private communication.

gress.